"Sylow-p Subgroups of Absolute Galois Groups, their Natural Quotients, and Galois Cohomology"

“绝对伽罗瓦群的 Sylow-p 子群、它们的自然商和伽罗瓦上同调”

• 批准号: 41981-2012
• 负责人: Minac, Jan
• 金额: $ 2.55万
• 依托单位: University of Western Ontario
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=593111
• 关键词: Sylow; Subgroups; Absolute; Galois; Groups

Almost 200 years ago E. Galois discovered a brilliant idea to study symmetries as one object, and in this way, to solve fundamental and seemingly intractible problems related to given mathematical structures. Today Galois theory is a central part of current mathematics. However some basic problems in Galois theory are still open. There are so many groups of symmetries that there became a desire to build enormous observing towers from which one could see all of these groups of symmetries. These towers are called absolute Galois groups. If one knew some special properties of absolute Galois groups, one would obtain a master key to many secret rooms of current mathematics. Remarkable progress was completed during the last 30 years by A. Merkurjev, M. Rost, A. Suslin, and V. Voevodsky, on obtaining very specific information in terms of cohomological invariants of absolute Galois groups. Thus we now have powerful information about absolute Galois groups, but it is encoded in the language of cohomology. The main objective of this proposed research is to decode some of the powerful information into down-to-earth descriptions of absolute Galois groups. In recent joint work with D. Benson, S. Chebolu, I. Efrat, J. Labute, N. Lemire and J. Swallow; we have found certain and various small quotients of absolute Galois groups. These groups can be viewed as foundations of our towers and now it is time to climb higher and to classify larger quotients of Galois groups. Based on joint work with M. Spira, and more recent work with A. Adem, D. Benson, I. Efrat, N. Lemire, A. Schultz and J. Swallow, on the interplay between topology, modular representation theory and the Galois module structure of Galois cohomology, we plan to provide further significant information about absolute Galois groups, and to apply it to the solution of basic problems in pure mathematics. Number theory and related topics in Canada are internationally well regarded. It is hoped that this project will contribute to sustaining this high standard and tradition in Canada.

大约200年前，E. Galois发现了一个绝妙的想法，将对称性研究为一个对象，通过这种方式解决了与给定的数学结构有关的基本且看似可怕的问题。今天，加洛瓦理论已成为当前数学的核心部分。但是，加洛伊斯理论中的一些基本问题仍然开放。有很多群体的对称性，以至于渴望建立巨大的观察塔，人们可以从中看到所有这些对称性。这些塔称为绝对Galois团体。如果一个人知道绝对Galois群体的一些特殊属性，那么人们将获得许多当前数学秘密房间的主钥匙。在过去的30年中，A。Merkurjev，M。Rost，A。Suslin和V. Voevodsky在过去的30年中取得了显着进展，以根据绝对Galois组的共同体不变性获得非常具体的信息。因此，我们现在拥有有关绝对Galois群体的强大信息，但是它是用同谋语言编码的。这项拟议的研究的主要目的是将一些强大的信息解码为绝对Galois组的脚踏实地描述。在与D. Benson，S。Chebolu，I。Efrat，J。Labute，N。Lemire和J. Swallow的最新联合合作中；我们发现了绝对Galois组的某些和各种小商。这些群体可以看作是我们塔楼的基础，现在是时候攀登更高并对Galois群体的更大商进行分类。基于与Spira M. Spira的联合合作，以及与A. Adem，D。Benson，I。Efrat，N。Lemire，A。Schultz和J. Swallow的最新工作，在拓扑，模块化表示理论和Galois之间的相互作用上GALOIS共同体的模块结构，我们计划提供有关绝对Galois组的进一步重要信息，并将其应用于纯数学中的基本问题的解决方案。加拿大的数字理论和相关主题在国际上受到了良好的评价。希望该项目能够在加拿大维持这种高标准和传统。